Silent introduction
Let's step back for a minute and take stock of the main message of what we've seen so far.
The purpose of looking at the logistic map
was to illustrate the phenomenon of chaos,
which means seemingly random behavior
with sensitive dependence on initial conditions.
The logistic map was a simple, completely deterministic equation
that, when iterated, can display this kind of seemingly random
behavior with sensitive dependence (depending on the value of R).
There was no randomness in the logistic map equation
Completely determined what the next value of x will be
by the previous value of x
And yet we see this seemingly random behavior
This is called deterministic chaos, where chaos arises from a
completely deterministic system or equation.
And the message to be learned here is that, if we have
deterministic chaos, perfect prediction, a la Laplace's deterministic
clockwork universe, which we saw earlier, is impossible,
even in principle, since we can't know
the precise value of the initial condition.
This is a profound negative result that, along with quantum mechanics, helped wipe out the optimistic 19th
century view of the clockwork Newtonian universe that kept going along in its predictable path in a deterministic way.
But what's the positive message?
Dynamical systems theory is the attempt to discover general principles concerning systems that change over time.
Through the work of many people and studying logistic map
and other similar deterministic equations, the result was to find some equally surprising profound and
positive results which is what we're going to call universality in chaos.
That is, characteristics that are universal across a certain broad set of chaotic systems.
In short, while chaotic systems are not predictable in detail,
there is a wide class of chaotic systems that have highly predictable universal properties.
Let's look at what these are.
Recall our logistic map bifurcation diagram,
which showed the period doubling route to chaos
And we had different, what we call, regimes of attractors. Remember we had our fixed point attractors?
And we had our periodic attractor regime.
And this led to a region of chaos where we have
what's called a chaotic attractor -- or what's also known as a strange attractor.
Note that there's some interesting looking structure in this chaotic attractor regime,
including places where the system goes back to being periodic.
Talking about that is really interesting, but it's beyond the scope of this particular course.
But there's something that is really surprising and interesting
and that is that this kind of period doubling route to chaos
that we see here also appears in many other chaotic systems.
The kinds of systems that "periodic route to doubling" appears in are what are called
the unimodal or one-humped maps.
Just like our logistic map that we're looking at right now,
you can see that there's a single hump in the parabola
And there are many such systems in nature that have this kind
of behaviour. They all have the same period doubling route
to chaos. Another map that shows that unimodal behaviour
is the so-called sine map.
It's called the sine map because it uses the trigonometric function of sine.
Here's what it looks like.
x sub t plus 1 = R/4 sin of (pi x sub t).
Now if you've forgotten trigonometry, don't worry.
Just think of this as a particular mathematical function that can
be iterated in the same way that we iterated the logistic map.
And notice here that the x doesn't stand for population anymore.
We're now in the realm of abstract mathematical functions
where x can stand for anything as long as it's between 0 and 1.
People often write this R/4 term as the Greek letter lambda.
I've left it as R over 4 just to keep the connection with the
previous notion of R in the logistic map.
But just think of this as a particular function
that we're going to iterate.
I've implemented the sine map as a NetLogo model
which you can download from the course materials website
And you can see that it also forms this parabola,
or one-hump map, as a logistic function
and it also displays the characteristic
period doubling route to chaos.
In the course materials site, it's called sinemap.nlogo
So now we've seen one universal property of chaos
in unimodal maps -- that is, the period doubling route to chaos
But there's another, even more surprising, universality that
was discovered in the 1970s and 1980s
by a few different groups of people.
I'll explain it by looking again at the logistic map bifurcation
diagram. Now in the 1970s, the physicist Mitchel Feigenbaum,
studied this map in great depth.
And he measured, as precisely as he could, the points
at which these various bifurcations occur.
And he found that the period two bifurcation appears
at about R = 3.0, period four at this value
period eight at this value
period sixteen at this value and so on
until finally, at a value of approximately 3.569946 followed by
some other decimal places, we have
a period of infinity. That is, we have chaos. It's aperiodic.
And that point is called "the onset of chaos."
Now if you notice, these bifurcations are coming quicker
and quicker as we vary R. So this goes from about .24 to .30
That's a long time. And then from .30 to .34 something,
that's shorter. And then shorter & then shorter & then shorter.
So what Feigenbaum did was he took these bifurcations
and he measured the rate at which the distance between
bifurcations is shrinking. The rate at which these are shrinking is just defined as the ratio of
this length
divided by the length to the next bifurcation
and then the ratio of this length
from the period two bifurcation to the period four bifurcation
divided by the length of the time from the period 4 bifurcation
to the period 8 bifurcation and so on.
Each of those ratios is a measure of the rate at which
the bifurcations are shrinking.
Let's take a look at these ratios mathematically.
This is what they look like.
The first ratio is the distance between the period 2 and
the period 4 bifurcations. That's R2 - R1
divided by the distance between the period 4 and the period 8 bifurcations.
Ok that gives us 4.7514 etc and that's our first estimate of
the rate at which the distance between the bifurcations is shrinking.
Ok. Then we can go to the next set and get the distance
between the period 4, period 8 divided by period 8, period 16.
That gives us 4.656 etc. And we keep doing that
And Feigenbaum did this using just a desktop calculator, and later a faster computer,
and found that, as we keep doing this -- as these Rs get bigger and bigger and bigger -- this number starts
converging to this value -- 4. 6692016
followed by some decimal places.
And in mathematical terms, that's the limit as n approaches infinity
of R n+1 minus R of n divided by R n+2 minus R of n+1.
But if you're not familiar with limits, don't worry about that.
Just look at these examples and you can see that,
as we increase the bifurcation number, we're going to get closer and closer to this value.
In other words, each new bifurcation appears about
this many times faster than the previous one.
That's what Feigenbaum found.
And this number has now become called Feigenbaum's constant
because, amazingly, not only did he derive this mathematically after he observed it
(he developed a whole theory that would show why this had to be true mathematically)
he also showed that any unimodal or one-humped map
like the logistic map or the sine map
will have the same value for this rate.
So this number is the universal constant for chaotic systems with one humped maps.
This number has also been experimentally observed
in a number of systems ranging from fluid flow to electric circuits, lasers, chemical reactions and so on.
Another amazing fact about this is that
at almost exactly the same time that Feigenbaum was carrying out his studies, the same thing was done by
another research team -- the French mathematicians Pierre Collet and Charles Tresser
They were completely independent of Feigenbaum
And this co-discovery is now sometimes called
The Feigenbaum, Collet, Tresser Theory.
In summary, let's look at what we've learned in this unit as to the significance of dynamics in chaos for complex systems.
Perhaps most importantly, we've seen an example
in which complex, unpredictable behavior arises from
very simple deterministic rules, such as the logistic map.
We've also seen that dynamical systems theory gives us a
vocabulary for describing complex behaviour using terms such as
"attractors" and "period doubling route to chaos" and so on.
We've seen that there are fundamental limits to detailed
prediction in systems with chaos due to "sensitive dependence on initial conditions."
At the same time, we've seen that there are universal properties of chaotic systems
and we might call this "order in chaos."